🔬 mesoscale physics

Zero-point energy of solids from vacuum fluctuation and quantum geometric force

This paper proposes that vacuum fluctuations of electromagnetic fields induce a volume-scaling zero-point energy in solids related to their quantum geometric properties, which can be experimentally detected through a measurable repulsive force or a static "quantum geometric force" acting on a superconducting LC circuit.

Original authors: Yugo Onishi, Liang Fu

Published 2026-02-10

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CC BY 4.0

Original authors: Yugo Onishi, Liang Fu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a room that looks completely empty. But according to the laws of quantum physics, that room isn't actually empty; it is buzzing with "invisible jitters"—tiny, restless waves of energy that never stop moving. This is what scientists call vacuum fluctuations.

Usually, these jitters are so small and subtle that they don't do much. But a new paper from MIT researchers suggests that if you place a specific kind of material (a solid) near a specialized electronic circuit, these tiny jitters can team up to create a real, measurable physical force.

Here is a breakdown of how this works using some everyday analogies.

1. The "Jittery" Circuit (The Source)

Think of a superconducting circuit (like an LC circuit) as a tiny, high-tech playground swing. Even when no one is pushing it, the swing is constantly vibrating slightly because of the "quantum jitters" in the air. These vibrations are the "vacuum fluctuations."

In a normal setting, these vibrations are harmless. But this circuit is special: it’s incredibly sensitive, like a swing that can feel the weight of a single dust mote.

2. The "Geometric" Material (The Target)

Now, imagine placing a solid object near this swing. The researchers aren't interested in just any object; they want materials with a special internal "map" called Quantum Geometry.

Think of the electrons inside a solid like people in a crowded ballroom.

In some materials, the people are arranged in a very rigid, predictable grid.
In "quantum geometric" materials, the way the people move and react to one another is incredibly complex and "curved." Their "dance moves" (how they shift when pushed) are dictated by the geometry of the room.

3. The Interaction: The "Invisible Hand"

When the "swing" (the circuit) jitters, it sends out tiny electromagnetic ripples. When these ripples hit the "ballroom" (the solid), the electrons inside try to dance along with them.

Because the electrons are so tightly packed and follow that complex "dance map" (the quantum geometry), their attempt to move actually pushes back on the circuit.

The result is two strange new forces:

The Repulsive Force: Imagine the circuit and the solid are two magnets that suddenly decide they hate each other. As they get closer, the "jitters" cause the solid to push the circuit away. It’s like a "Quantum Ghost" pushing the circuit back.
The Geometric Force: Because the circuit's ability to vibrate depends on its internal parts (its capacitor), the material actually pulls on the circuit's components. It’s like the ballroom dancers are so synchronized that their dancing actually tugs on the walls of the room.

Why does this matter?

For a long time, "Quantum Geometry" was a theoretical concept—a beautiful mathematical map that scientists thought was almost impossible to "touch" or measure directly. It was like knowing a mountain exists because of a map, but never being able to actually climb it.

This paper provides a "Quantum Measuring Stick." By building these circuits, scientists can actually feel the geometry of the electrons. If the force is strong, the "dance map" is complex; if the force is weak, the map is simple.

In short: The researchers have found a way to turn the invisible, restless "jitters" of empty space into a mechanical tool that allows us to feel the hidden, geometric architecture of the microscopic world.

This paper, titled "Zero-point energy of solids from vacuum fluctuation and quantum geometric force" by Yugo Onishi and Liang Fu, proposes a novel macroscopic quantum effect where the vacuum fluctuations of an electromagnetic field induce a volume-extensive zero-point energy in solids, which can be measured via a new type of force.

1. The Problem

Traditional quantum electrodynamics (QED) effects, such as the Lamb shift (energy level shifts in atoms) and the Casimir effect (attractive forces between plates), arise from vacuum fluctuations. However, these are typically single-particle effects or result in attractive forces.

The authors address a gap in understanding how these fluctuations interact with macroscopic many-body systems (solids with $\sim 10^{23}$ electrons). Specifically, they investigate how the coupling between a superconducting circuit (a controlled source of vacuum fluctuations) and a quantum material can reveal fundamental, intrinsic properties of the material's ground state—specifically its quantum geometry.

2. Methodology

The authors employ an adiabatic approximation to study the interaction between a superconducting LC circuit and a gapped electronic system.

System Modeling: The total Hamiltonian $H_{tot}$ is composed of the electronic system $H[\hat{A}, \hat{\phi}]$ and the superconducting LC circuit $H_p$ . The circuit provides fluctuating vector ( $\hat{A}$ ) and scalar ( $\hat{\phi}$ ) potentials.
Adiabatic Framework: Because the electronic energy gap $\Delta$ is much larger than the microwave photon energy ( $\hbar\omega_p$ ), the electronic system is "fast" and remains in its instantaneous ground state.
Unitary Transformation: They use a unitary transformation to diagonalize the Hamiltonian, allowing them to derive an effective Hamiltonian $H_{eff}$ that accounts for the entanglement between the electrons and the virtual photons.
Geometric Derivation: They relate the resulting energy shift to the many-body quantum metric ( $G_{ij}$ ), a fundamental geometric property of the ground state wavefunction.

3. Key Contributions

Identification of $E_{QG}$ : They derive a new term in the ground state energy, $E_{QG}$ , which is the quantum geometric zero-point energy.
The "Quantum Weight" Connection: They show that for insulators, this energy density is proportional to the quantum weight ( $K_{ij}$ ), an intensive property characterizing long-wavelength density fluctuations.
Definition of "Quantum Geometric Force": They define two distinct manifestations of this energy:
1. A repulsive force between a dielectric and an inductively coupled resonator.
2. An attractive force acting directly on the capacitor plates of the circuit.
Distinction from Casimir/Lamb Effects: They mathematically demonstrate that while the Lamb shift is negative and the Casimir force is typically attractive, the $E_{QG}$ shift is positive and the resulting force can be repulsive, due to its origin in diamagnetic-like inductive coupling.

4. Key Results

Energy Scaling: Unlike single-atom effects, $E_{QG}$ is extensive; it scales with the volume $\Omega$ of the solid.
Mathematical Expressions:
- The energy shift: $E_{QG} = \frac{e^2}{2C_p} g G g$ .
- The force on the capacitor: $F_{QG} = \frac{e^2 V}{4\pi\epsilon S} (g_I K' g_I + 2(\nabla g_C) K' g_I)$ , where $K'$ is the quantum weight.
Experimental Feasibility: The authors provide order-of-magnitude estimates. For a $1\text{ mm}^3$ volume, they predict $E_{QG} \sim 0.1\text{ eV}$ and a force $F_{QG} \sim 10\text{ pN}$ . These values are well within the detection limits of modern microelectromechanical systems (MEMS) and SQUID-on-lever techniques.
Robustness: The effect is a ground-state phenomenon, meaning it persists even in the absence of real photons (vacuum state) and is largely independent of temperature, provided $T$ is below the electronic gap $\Delta$ .

5. Significance

This work is significant for several reasons:

New Probe for Quantum Materials: It provides a direct, static, and macroscopic method to measure the many-body quantum metric and quantum weight. This is a major advantage over previous methods that required complex sum rules or broad-frequency optical measurements.
Bridging QED and Condensed Matter: It establishes a new regime of "Circuit QED" where the interaction is not just with artificial atoms (qubits) but with the collective geometric properties of bulk matter.
Topological Physics: Since the quantum weight is bounded by topological invariants (like the Chern number), this force could serve as a tool to detect and characterize topological phase transitions in Chern insulators and other quantum materials.